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EMPOWERR SUMMER 2009 WORKSHOP
MATHEMATICS ACTIVITIES
FRACTIONS AND ADDITION
We recall our definition of fractions in the box below.
THE NUMBER LINE DEFINITION OF FRACTIONS
We take as our starting object the number line with the whole numbers marked off
as dots on it. We also assume the following axiom of the number line to be true.
AXIOM OF THE NUMBER LINE: Given any whole number l with l ≠ 0 and given any
segment of the number line, it is possible to split that entire segment up into
exactly l pieces, each one of which has the same length as each and every one of all
the other pieces.
Definition.
Let k, l be whole numbers with l > 0. Divide each of the line segments
[ 0, 1], [ 1, 2], [2, 3], [3, 4], . . . into l segments of equal length (this is possible
because of the axiom of the number line given above).
These division points together with the whole numbers now form an infinite
sequence of equally spaced dots on the number line (in the sense that the lengths
of the segments between consecutive dots are equal to each other).
The first dot to the right of 0 is labeled
1
.
l
SO THE DEFINITION OF THE SYMBOL
1
l
IS THAT IT IS THAT DOT- it is the first one to the right of 0.
The second dot to the right of 0 is by definition
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2
3
k
, the third , etc., and the k-th one is
.
l
l
l
1
Fractions in lowest form and GCF
We know that any fraction can be expressed in many different ways (in fact, in
infinitely many different-looking ways).
For example, we have the equalities
12 / 50 = 6 / 25 = 30 / 125 = 54 / 225.
In all these different ways of labeling the same dot on the number line, only 6 / 25
has the property that the numerator and denominator have no common factors
except one.
For example
for 12/50 one has GCF (12, 50) = 2
for 6/25 one has GCF(6, 25) = 1
for 30 / 125 one has GCF(30, 125) = 5
for 54 / 225 one has GCF(54, 225) = 9
Definition If a and b are whole numbers with b ≠ 0 then the fraction a/b is said to
be in lowest form if GCF ( a , b ) = 1, i.e. a and b are relatively prime (that is, a and
b have no common prime factors).
In general let x and y be natural numbers and consider the fraction x/y. You can
give a formula for the correct expression for x/y in lowest form or simplest form
using a combination of x, y, and GCF(x,y)
x/y in lowest form is given by [ x / GCF(x,y ) ]
/ [ y / GCF(x,y ) ]
CHALLENGE QUESTION:
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Some students might say that when a and b are relatively prime, the fraction a/b
cannot be “reduced any further”, because there are no common factors for a and b
except 1 (why are there no other common factors?) so there is no ‘canceling out’
that can occur.
Can you explain the idea of reducing a fraction in terms of the number line model
of fractions we have adopted? For example, what does it mean in terms of the
number line definition of fractions, that 30 / 125 “can be reduced” to 6 / 25, but
that it cannot be reduced ‘any further’ ?
Can you give a way to think of the lowest form of a fraction in terms of the number
line definition of fractions?
ADDITION OF FRACTIONS
Another advantage of the number line definition of fractions is that we can extend
the operations of addition (and subtraction) from whole numbers to fractions very
easily. We can define addition and subtraction for whole numbers using the number
line.
We can then extend addition to fractions using the same ideas. Each fraction is a
dot on the number line. Each dot on the number line has associated to it an arrow
pointing from 0 to that dot.
Whole number addition on the number line is described by arrow concatenation or
juxtaposition as we now describe.
If a and b are whole numbers and we associate to a and b the corresponding
number line arrows, then a + b is the dot on the number line obtained as follows.
Take the initial point of the b arrow and place it at the terminal point of the a
arrow. This newly placed b arrow points to a new number and we define this new
number to be a + b.
In the box below we look at the example of 3 + 5 via the number line.
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3
|---------------------------->
5
|---------------------------------------------->
+--------+--------+--------+--------+--------+--------+--------+--------+--------+--…
0
1
2
3
4
5
6
7
8
9
3
5 (started at the 3 endpoint)
|---------------------------->|--------------------------------------------->
8
|---------------------------------------------------------------------------->
+--------+--------+--------+--------+--------+--------+--------+--------+--------+---…
0
1
2
3
4
5
6
7
8
9
We use exactly the same idea to define fraction addition: each fraction is a dot on
the number line, so has an arrow associated to it. To add one fraction to another,
just juxtapose the arrows in the order like the one for whole numbers. The second
addend will point to a new dot on the number line, and that dot is defined to be the
sum of the two original fractions.
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Example: 3/5 + 2/3
3/5
|-------------------------
|--------+--------+--------+--------+--------|--------+--------+--------+--------+------|---------
1
2
0
1/5
2/5
3/5
4/5
5/5
6/5
7/5
8/5
9/5 10/5
2/3
|----------------------------
|--------------+--------------+---------------|--------------+--------------+---------------|----------
1
2
0
1/3
2/3
3/3
4/3
5/3
6/3
3/5
2/3
|-------------------------|---------------------------- = 3/5 + 2/3
|--------+--------+--------+--------+-------|--------+-------+--------+--------+------|---------
1
2
0
1/5
2/5
3/5
4/5 5/5
6/5
7/5
8/5
9/5 10/5
So 3/5 + 2/3 gives a new dot on the number line-it is the dot between
6/5 and 7/5.
But how do you know 3/5 + 2/3 is a fraction according to our definition of
fraction? In other words, can you give a reason why that dot is the address or
location of a point on the number line, obtained by dividing each unit interval up
into a certain whole number of pieces, then counting over a whole number of those
pieces from 0?
In more mathematical-sounding language, we might ask this: how do we know that
the number line definition of fractions is closed under this definition of fraction
addition?
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